[leftmargin=+4mm]
(a)
Epitaxially strained thin films, e.g.,
[1]}, [2]}, [3]}, [4]}, [5]}, [6]}:
\(\Omega :=(a,b)\times (0,+\infty ),\) 
\(S:=(a,b)\times (-\infty ,0)\) 
for some \(a<b,\) 
free crystals in the subfamily
\(\begin{aligned}\quad \mathcal {A}_{\rm subgraph}:=\lbrace A\subset \Omega :\,\text{$\exists h\in BV(\Sigma ;[0,\infty ))$ andl.s.c. such that $A=A_h$} \rbrace \subset \mathcal {A}_1,\end{aligned}\) 
where \(A_h:=\lbrace (x^1,x^2)\,:\, 0<x^2<h(x^1)\rbrace \) ,
and admissible configurations in the subspace
\(\mathcal {C}_{\rm subgraph}:=\lbrace (A,u):\, A\in \mathcal {A}_{\rm subgraph},\,u\in H_\mathrm {loc}^1({\rm Int}{(A\cup S\cup \Sigma )};\mathbb {R}^2)\rbrace \subset \mathcal {C}_1\) 
(see also [7]}, [8]});
(b)
Crystal cavities, e.g., [9]}, [10]}, [11]}, [12]}:
\(\Omega \subset \mathbb {R}^2\)  smooth set containing the origin,
\(S:=\mathbb {R}^2\setminus \Omega ,\)  free crystals in the subfamily
\(\mathcal {A}_{\rm starshaped}:=\lbrace A\subset \Omega :\,\text{open, starshaped w.r.t. $(0,0)$,and $\partial \Omega \subset \partial A$} \rbrace \subset \mathcal {A}_1,\) 
and the space of admissible configurations
\(\mathcal {C}_{\rm starshaped}:=\lbrace (A,u):\, \, A\in \mathcal {A}_{\rm starshaped},\,u\in H_\mathrm {loc}^1({\rm Int}{(A\cup S\cup \Sigma )};\mathbb {R}^2)\rbrace \subset \mathcal {C}_1;\) 
(b)
Capillarity droplets, e.g., [13]}, [14]}, [15]}:
\(\Omega \subset \mathbb {R}^2\)  is a bounded Lipschitz open set (or a cylinder),
admissible configurations in the collection
\(\mathcal {C}_{\rm capillarity}:=\lbrace (A,u_0):\, \,A\in \mathcal {A}\rbrace \subset \mathcal {C}\quad \text{or}\quad \color {black}\widetilde{\mathcal {C}}_{\rm capillarity}:=\lbrace (A,u_0):\, \,A\in \widetilde{\mathcal {A}}\rbrace \subset \widetilde{\mathcal {C}};\) 
(d)
Griffith fracture model, e.g., [16]}, [17]}, [18]}, [19]}, [20]}, [21]}, [22]}:
\(S=\Sigma =\emptyset \) , \(E_0\equiv 0\) , and
the space of configurations
\(\mathcal {C}_{\rm Griffith}:=\lbrace (\Omega \setminus K,u):\,\text{$K$ closed, $\mathcal {H}^1$-rectifiable},\,u\in H_\mathrm {loc}^1(\Omega \setminus K;\mathbb {R}^2)\rbrace \subset \mathcal {C};\) 
(e)
Mumford-Shah model, e.g., [23]}, [24]}, [25]}:
\(S=\Sigma =\emptyset ,\)  \(E_0=0,\) 
\( is such that the elastic energy \) W\( reducesto the Dirichlet energy, andthe space of configurations\)\(\mathcal {C}_{\rm Mumfard-Shah}: =\lbrace (\Omega \setminus K,u)\in \mathcal {C}_{\rm Griffith}:\,\,u=(u_1,0)\rbrace \subset \mathcal {C};\)\(\)
(f)
Boundary delaminations, e.g.,
[26]}, [27]}, [28]}, [29]}, [30]}, [31]}:
the SDRI model includes also the setting of
debonding and edge delamination in composites [31]}. The focus is here on the 2-dimensional film and substrate vertical section, while in [26]}, [29]}, [30]} a reduced model for the horizontal interface between the film and the substrate is derived.

